Scaling law and asymptotic distribution of Lyapunov exponents in conservative dynamical systems with many degrees of freedom

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Abstract
The authors study by numerical means the infinite product of 2N*2N conservative random matrices which mimics the chaotic behaviour of Hamiltonian systems with N+1 degrees of freedom made of weakly nearest-neighbour coupled oscillators. The maximum Lyapunov exponent lambda 1 exhibits a power-law behaviour as a function of the coupling constant epsilon : lambda 1 approximately epsilon beta with either beta =1/2 or beta =2/3, depending on the probability distribution of the matrix elements. These power laws do not depend on N and moreover increasing N, lambda 1 rapidly tends to an asymptotic value lambda *1 which only depends on epsilon and on the kind of probability distribution chosen for building up the matrices. They also compute the spectrum of the Lyapunov exponents and show that is has a thermodynamic limit of large N. This suggests the existence of a Kolmogorov entropy per degree of freedom proportional to lambda *1.